Invariants
| Base field: | $\F_{7}$ |
| Dimension: | $3$ |
| L-polynomial: | $( 1 - 5 x + 7 x^{2} )( 1 - 3 x + 7 x^{2} )^{2}$ |
| $1 - 11 x + 60 x^{2} - 199 x^{3} + 420 x^{4} - 539 x^{5} + 343 x^{6}$ | |
| Frobenius angles: | $\pm0.106147807505$, $\pm0.308124534521$, $\pm0.308124534521$ |
| Angle rank: | $2$ (numerical) |
This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable.
Newton polygon
This isogeny class is ordinary.
| $p$-rank: | $3$ |
| Slopes: | $[0, 0, 0, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
|---|---|---|---|---|---|
| $A(\F_{q^r})$ | $75$ | $117975$ | $46785600$ | $14572861875$ | $4736816720625$ |
Point counts of the (virtual) curve
| $r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
|---|---|---|---|---|---|---|---|---|---|---|
| $C(\F_{q^r})$ | $-3$ | $49$ | $396$ | $2525$ | $16767$ | $116716$ | $821601$ | $5768021$ | $40385412$ | $282573289$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{7}$.
Endomorphism algebra over $\F_{7}$The isogeny class factors as 1.7.af $\times$ 1.7.ad 2 and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
Base change
This is a primitive isogeny class.